JPS638880A - Free curved surface forming method - Google Patents

Abstract

PURPOSE:To form an always smooth curved surface by selecting the changing ratio of a cross sectional tangent vector to a border curved line to surround a framing space to a prescribed value and sticking a smooth patch to the framing space. CONSTITUTION:By sticking an S(u,v) displayed by a vector function having a parameter to show a position to a framing space surrounded by border curved lines BOD1-BOD4, a free curved surface is generated. Here, internal control points P(11)-P(22) corresponding to respective border curved lines BOD1-BOD4 are set by cross sectional tangent vectors (a01,a02) and (a21,a22) to respective border curved lines BOD1-BOD4. By selecting the changing ratio in the direction along the corresponding border curved line of the cross sectional tangent vector to a prescribed value, the position the control points P(11)-P(22) are not crossed each other, overlapped and made approximate, and the overturning and folds will not be generated at a patch S(u,v) stuck to the framing space.

Description

[Detailed description of the invention] The present invention will be explained in the following order.

A. Industrial field of application B. Outline of the invention C. Conventional technology Problems to be solved by the invention (Fig. 6 (A) and (
B) Means for solving E problems (Figs. 1 to 5) F effects (Figs. 1 to 5) G embodiment (Gl) 1st embodiment (Figs. 1 and 2) (G2) Second
Example (Figures 3 and 4) (G3) 3rd example (5th example)
(G4) Other embodiments H Effects of the invention
computer aided design), or CAM (c.

mputer aided manufacturer
In g), etc., this method is suitable for use when generating a shape with an inwardly curved surface.

B. Summary of the Invention The present invention provides a free-form surface creation method that generates a free-form surface by extending patches represented by predetermined vector functions in a framework space, in which a transverse tangent vector to a boundary curve surrounding the framework space is calculated. By selecting the rate of change to a predetermined value, a smooth patch can be spread over the framework space.

C Conventional technology For example, when designing the shape of an object with a free-form surface using a CAD method (gio metric mode)
ling), the designer generally specifies multiple points in three-dimensional space (these are called nodes) that the curved surface should pass through,
A boundary curve network connecting the specified plurality of nodes is calculated by a computer using a predetermined vector function, thereby creating a curved surface expressed in a so-called wire frame. In this way, a large number of framework spaces surrounded by boundary curves can be formed (such processing is called framework processing).

The boundary curve network formed by such framework processing itself represents the rough shape that the designer intends to design, and a curved surface that can be expressed by a predetermined vector function is calculated using interpolation using the boundary curves surrounding each framework space. If it is possible to do so, it is possible to generate a free-form surface (which cannot be defined by a quadratic function) that is designed by the designer as a whole. Here, the curved surfaces stretched across each framework space form basic elements constituting the entire curved surface, and these are called patches.

Conventionally, in this type of CAD system, easy-to-calculate vector functions expressing the boundary curve network are used, such as the Bezier equation and B-spline (B-apli).
A third-order tensor product formed by the equation (ns) is used, and is considered to be optimal for mathematically expressing a free-form surface that does not have any special features in terms of shape, for example.

In other words, a free-form surface that has no special features in shape is
When points given in space are projected onto the xy plane, the projected points are often arranged regularly in a matrix, and when the number of projected points is expressed as mXn, the framework space can be easily stretched using a quadrilateral patch expressed by a cubic Bezier equation.

D. Problems to be Solved by the Invention When a quadrilateral framework space expressed by a cubic Bezier equation is formed by such framework processing, the interior of each quadrilateral framework space is generally a vector function of any degree. It is thought that the curved surface expressed by can be stretched.

However, in reality, when the degree of the vector function representing a quadrilateral patch increases, the four surrounding edges are defined by the framework processing, so the internal Depending on the position of the control point, there is a possibility that the stretched patch may have a partially uneven shape. Therefore, if you want to stretch a rectangular patch with a curved surface as smooth as possible to obtain a natural feeling, use this method. There is a risk that it will be difficult to realize this.

As a way to solve this problem, the internal control points are determined using a conditional expression in which the degree is lowered by dividing the vector function representing the patch to be extended in the framework space into a partial differential with respect to the parameters U and V. A method has been proposed (Japanese Patent Application No. 61-64560).

However, even with this method, depending on the shape of the quadrilateral framework space formed by the framework processing, internal control points Pttl, P3. ) are set at positions where they intersect (Fig. 6 (A)), overlap each other (Fig. 6 (B)), or are set at positions close to each other, the generated quadrilateral patch There is a risk that parts of the paper may turn inside out or become severely distorted, causing wrinkles.

In addition, in FIG. 6, P. . 1,P(,.1,P(33
1.P. 1. represents the nodes formed by framework processing, and P(1@), P(!11>, P(ffl+
and PBz+, P 1! and Pars>, P(0 and pHll> are the boundary curves BO surrounding the framework space, respectively)
Represents a control point consisting of control edge vectors representing DI, BOD2, BOD5, BOD4, and further includes P5.1, P++z+
, P(11), P(t. represents the internal control point for applying the patch.

The present invention has been made in consideration of the above points, and is a free-form surface that can always generate a smooth curved surface as a patch to be stretched in a framework space without causing such flipping or wrinkles. This is an attempt to propose a method of creation.

E Means for Solving Problem E In order to solve this problem, in the present invention, a large number of framework spaces surrounded by boundary curves BOD1 to BODA are formed by framework processing, and parameters representing positions are set in the framework space. A patch S represented by a vector function with
In a free-form surface creation method that generates a free-form surface by stretching Tu+ITl, internal control points P, 8.corresponding to each boundary curve 311BOD1 to BODA. ~P, zt+ as each boundary curve BODI−
Transverse tangent vector (aol, a, ri,
(ax a it), (boo, b to), (bus
, b ts), and this transverse tangent vector (aS a ox), (311%
M! ri, (bus-bto), (bus, b-yo,
) by selecting the rate of change in the direction along the corresponding boundary curve to a predetermined value, the internal control point P(lθ~P(.,
Avoid intersecting, overlapping, or close locations.

Transverse tangent vectors corresponding to F-action boundary curves BOD1 to BOD4 Caors aoz), (atls atx), (b
l. By making it possible to control the rate of change of , b8゜), (bll, bltsu), the transverse tangent vector (aols aat) 5(atls"!ris(
internal control points P(II) to P, represented by bus, b8°), (bl3, bts); , intersect the positions of ,
By preventing the patches from overlapping or coming close to each other, it is possible to prevent the patches stretched in the framework space from turning over or wrinkling.

G example An embodiment of the present invention will be described in detail below with reference to the drawings.

(G1) First Embodiment In the first embodiment, the boundary curve representing the boundary of the quadrilateral framework space subjected to framework processing, and the patch S (un vl) stretched in each quadrilateral framework space are expressed as It is expressed using a vector function consisting of a third-order Bezier equation, as shown in Fig. 1. In equation (1), P, .
(node P(..2
,P. 2.P. 32, P. , the position vector represents the reference position.

Thus, the quadrilateral patch S (III Ill is the node P+aa>-P +5u-P t3s>-P (.5)
-P <au's four boundary songs! IIBODI, BOD2
, BOD5, and BOD4. Each of these boundary curves defines a cubic Bezier equation using two control points.

That is, node P,. . , and P(3゜1, Pf3゜,
and P O3), P tsn and P(on, P(031
and P (061 between the boundary curves BODI, BOD2, B
OD5 and BOD4 each have two control points (PLl,
), P(101)S (Push・P+sg+)・(
P(寞1)・P＜+a))・(P(.2.,P,.1)
).

In addition, in equation (1), E and F are shift operators in the U direction and V direction, and for the control point P (□′14.) represented by the position vector on the patch S (un wt EP (Ei, j, = p, 5 and 1.1) (i + j =
0.1.2)... (2) FP, Direct, C ==
P (Direct・j・1)(1,j−0,1,2)...
... (3) It has the following relationship.

Further, U and V are parameters representing positions in the U direction and V direction, and change between 0 and 1 as expressed by the following formula % formula % (4). Thus, as shown in FIG. 1, the node P,. . , the coordinates on the free-form surface in the patch S (un v) can be expressed using coordinates (U, V) with the U axis taken in the horizontal direction and the V axis taken in the vertical direction.

When defined in this way, two control points (P(□1, P, .8
．． ), (Peso, P<sg,), the set of control points P(Ill, P(1!I) and P4°, P(ffi.

By setting the set of four boundary curves BOD1, BOD2, BOD defined by framework processing,
5. A free-form surface defined by four internal control points P(Ill to P(IH) can be stretched in the quadrilateral framework space surrounded by BOD4.

Here, the internal control points P (Ill and PHx) are the corresponding control points P, respectively. 0, the corresponding boundary song with P(.t) as the starting point, v! It is set on the B OD d at a position represented by a transverse tangent vector al11 and aoz pointing in a direction that crosses it (that is, the U direction).

Similarly, the internal control points P(!11 and P (tt) are from the corresponding control points P.1. and PL3°,
A transverse tangent vector a! on the boundary curve BOD2 in a direction that crosses it (that is, the U direction)! I and ao
is set to the position represented by .

Here, the transverse tangent vector al11 and a,,2 are expressed by the following formula a.3+2ao.

......(5) As expressed by the following, the control side vector in the U direction is set at 3 degrees during framework processing. Control side vector a6 in the V direction from
The rate of change is selected so that it remains constant up to 3.

Similarly, the transverse tangent vectors att and a. is set to change at a constant rate of change from the control side vector a2° to the control side vector a in the V direction, as shown in the following equation.

Here, the control edge vector a,. and a(13, ago and a, respectively, are the boundary curves BOD1 and Control points P(He) and P(Ill, P(!0
1 and P(!), thus defining the boundary curves BOD1 and BOD5.

(5) The transverse tangent vector all and a *ts a! By selecting l and at□, the internal control points P(Ill and P(
10, by setting PH1+, P, zzr, these internal control points pan and P(Ill, P(211 and p
(z□ will not intersect, overlap, or be very close to each other, and will always be set at positions with appropriate spacing.

This can be confirmed as follows. When the patch stretched in the framework space is expressed using the free-form surface equation of equation (1),
The tangent vector in the U direction at each point of the patch is calculated using the following formula: %=3(1-u+uE)"(1-v+vF)"aaa...
...... As shown in (9), it can be expressed by one-time partial differentiation of the parameter U, and therefore, the transverse tangent vector on the boundary curve BODd can be expressed as the following equation by placing 11 z Q in equation (9). Equation % (10) On the other hand, the rate of change in the tangent vector in the U direction at each point of the patch can be expressed as the following equation by partially differentiating equation (9) once with respect to the V direction. , Therefore, the rate of change of the transverse tangent vector on the boundary curve BOD4 is given by Equation (11) =
0, the following formula% formula%) Here, expand the square term on the right side of formula (12), and (
F-1) a,,, F (F-1) aao, F”
(F-1) ao. (F 1)aoo-ao+-aoo=Bo.

・・・・・・(13) F(F-1)aoo=aoz-ao+-FBoo=Bo
t...(14) F" (F-1) aoo-aoi-aoz=F"Boo=
Boz... (15) The difference vector B, as shown in (15). , if you put BoiB11,
The formula (12) is the following formula = 9 ((1-VsB60 +2 (I V)VB
III+v”Bot) ・・・・・・(1
It can be transformed as shown in 6). So the difference vector B, IB
If oB, 2 is set to the same value 13o+t as shown in the following formula (% formula %), then the rate of change of the transverse tangent vector can be expressed by the reference difference vector B, as shown in the following formula (% formula %) Can be done. This means that a change in direction can be represented by a constant vector.

Equation (18) shows that the rate of change of the transverse tangent vector, that is, the change from the control side vector a0° to the control side vector aos through the transverse tangent vector a61 and aoz sequentially along the boundary curve BOD4, is expressed as shown in FIG. As shown in , it shows that the reference difference vector BOW changes in the same direction and size, and this means that in equation (17), equation (13), equation (14), and The following formula obtained by substituting the formula all aoo” aaa-a
ll”B63aaa”B11k゛・・・・・・(
19).

Therefore, in equation (19), the reference difference vector B6. is the control edge vector a, given during framework processing. and a63
In order to express it by, the first equation of equation (19) all
-a@2nd formula a@! -a@3rd formula a0. -a,! If we add them together and take the average, we get 3 Bam=(aOt-
aoa) + (aaz-aOt)+(aai-aaz
) -aaz-aoo ・・・・・・(20)
Bam= (ao3aoo) ・・”” (
21). Therefore, by substituting equation (21) into equation (17), we obtain the difference vector B. and Bol, B11+1= aOt-aoa-(ao3-aoo)
......(22) Ba snow-aox Bam= (aos-aoo
)... (23) From these equations (22) and (23), the transverse tangent vectors aot and ant can be expressed by the known control side vectors a0° and aoi as shown in the following equations. .

Thus the transverse tangent vectors aol and a. As t, (
If the relationship is set as expressed by equations (24) and (25), then the transverse tangent vectors al11 and ao
The internal control points P(,) and PCll represented by x
l is the control edge vector a, as shown in FIG. and aos are set at such a position that the difference between the reference difference vectors B and K is changed at a constant ratio. As a result, it is effective to form control points P, ) and PH+t+ set by transverse tangent vectors al11 and 3.2 at positions where they intersect, overlap, or are very close to each other. can be prevented.

Equations (13) to (25) describe how to set the transverse tangent vectors al11 and aOt representing the control points P(Ill and P(It) corresponding to the boundary curve BOD4. Similarly, for WBOD2, the transversal tangent vectors specifying the control points PH1+ and P(Ill) are calculated using the following formula % formula % without intersecting with each other, overlapping with each other, or approaching the vicinity of scratching. In the case of the example, the internal control point P(Ill
and P(,), control points P(Ill) and P1oz+, P(31) and P of boundary curves BOD4 and BOD2 extending PC211 and PBz+ to face each other in the V direction
(I mentioned the case where you set it from 3□, but
Instead, as shown in FIG. Points PH1, and P(zn, P(1
! Transverse tangent vectors b1° and b representing l and P, 2°
In order to have a constant relationship between the change rates of b13 and bt3, the control side vector b given during framework processing is calculated as shown in the following equation.
,. and b3°, b03 and b13.

In this case, the formula ((1
) by the parameter V, the tangent vector aS (u, v) V = 3(1-v+vF)''(1-u+uE)'be in the V direction.

(32) By finding the following and further partially differentiating it with respect to the parameter U, the rate of change of the tangent vector in the U direction a''s, .

vau = 9(1-v+vF)"(1-u+uE)"(E-1)
t),. ...-(33) can be obtained. By setting v=Q in equation (33), the rate of change of the transverse tangent vector=9(1-u+uE
)”(E-1)bo.

・・・・・・(34) can be found.

The difference vector obtained by expanding equation (34) is set as the difference vector AoosA1A z 6 as shown in the following equation, and A. -A1゜=A2゜=A, 1...
・As shown in (38), the reference vector A is mutual. ,
For the difference vector, b+o-boo=bzo b+o=b3o bzo
``Aog... (39) The relationship holds true, so that the reference vector A, 1 can be expressed using the control side vector b0° and the following equation (40). Can be done.

Therefore, from equations (35), (37), and (40), A
,. -S. -bo. ---(bko.-bo.)...
... (41) A. -S. -b8゜” (bs.-bo.

)...... (42) The relationship can be found, and as a result, the boundary curve BOD1
The transverse tangent vectors blO and bto for can be determined by .

Similarly, the transverse tangent vectors b13 and bs3 corresponding to the boundary curve BOD5 can be obtained as follows.

In this way, from the relationship in equation (39), the control side vector b0 including the transverse tangent vectors bi11 and b2°
゜Crow. The rate of change up to the reference vector A is shown in FIG. By being constant by K, the internal control points P(Ill and P(!l) set by the transverse tangent vectors blO and b2° are set at positions that maintain appropriate spacing, so they do not intersect with each other. It is possible to effectively avoid overlapping, overlapping, and approaching the scratching position.

(G3) Third Embodiment FIG. 5 shows a third embodiment, and as shown in FIG. 5 corresponding to FIGS. 2 and 4, the difference vector B0 and B
As shown in the following formula, where B and t are scalar quantities of arbitrary values,
, h2, h, multiplied by the reference vector BOK.

In this way, the difference vector B0゜, Bo]Ba
t is a coefficient in the direction of the reference vector Bow, h! ,h
, becomes a position vector representing a position that changes from an equal interval by the magnitude determined by .

Here, the interval between the difference vectors is calculated by taking into account the limit of the allowable free-form surface that is currently being created, and h
If t and h are arbitrarily selected, the internal control point P(11)
and Po. The distance between them can be set to such an interval that practically no distortion such as turning over or wrinkles occurs in the generated patch.

In this case, the transverse tangent vectors a61 and a,
! can be found as follows.

First, the difference between the control side vectors a06 and ass is expressed as the sum of the difference vectors B0°, BOI, B, , (47), (48
) and (49), the following formula Boo+Bo++Box=(hl+hz+h3)Bat
・・・・・・(50) In addition, from the above equations (13), (14), and (15), the following equation B+6 + B(I
I + Box-(a (lI-a oo) + (aog
-aa+)+(aoz adz)-3, 3-3,.・
...It can be obtained as in (51).

Therefore, from equations (50) and (51), the reference difference vector B
@, is as follows, and the difference vector B0° and 13oz including the transversal tangent vector all and 86K to be found is calculated by the following formula %
It can be expressed by a known control edge vector aO(1 and a., coefficients h+, hz, hs) as shown in the formula.

Thus, from equations (53) and (54), the transverse tangent vectors a01 and a. 2 beams.

h, +h! It can be found from

Even in this case, the internal control points P(II) and P(lt
) can be effectively avoided from intersecting with each other, overlapping, or placing them in very close positions.

(G4) Other embodiments (11 In the embodiment of FIG. 5, the boundary curve BOD4 (
transverse tangent vectors al11 and an
Although the case where t is set has been described, the present invention is not limited to this, and can be similarly applied to a case where internal control points are set based on the boundary curves BOD2, BODI, and BOD5.

(2) In the above, the transverse tangent vectors (a, and aoz, az+ and a,) (Fig. 1), (blo and bto, b13 and b) are separately expressed in the U direction or the V direction.
o) (Fig. 3), but instead of this, each boundary curve BODI, BOD2,
For each control point of BOD5 and BOD4, two corresponding control points are found separately, and the internal control points are expressed using a so-called superposition expression method in which one patch is represented by overlapping all of these control points. The present invention can also be applied to a method for creating a curved surface in which .

(3) Furthermore, in the embodiments of FIGS. 2, 4, and 5, the transverse tangent vectors a, G02, G2, and aZ
2z The positions represented by b16, bB, b13, and bo are converted into control edge vectors alIO, aol, ago, and a! 3.b
o.

and. , bll and s. has been described above, the present invention is not limited to this, and the control side vectors a0° and al13, ate
and a! 3, b (Il+ and bs., bos and b31
A transversal connection vector a0 is placed on a predetermined curve connecting the points represented by
1 and a. 2, or ail and G12, or bllll and b2°, or b1u and bzs, the same effect as in the above case can be obtained.

H Effects of the Invention As described above, according to the present invention, the positions represented by the transverse tangent vectors to be set based on the control points defining the boundary curve are selected so as to maintain a predetermined interval. It is possible to easily realize a free-form surface creation method that effectively avoids the possibility that the stretched patch will be turned inside out or wrinkled.

[Brief explanation of the drawing]

FIG. 1 is a route map showing a patch in the first embodiment of the free-form surface creation method according to the present invention, FIG. 2 is a route map showing the relationship between the transverse tangent vector and the reference difference vector, and FIG. Route map showing patches in the second embodiment, No. 4
The figure is a route map showing the relationship between the transverse tangent vector and the reference difference vector, and FIG. 5 is a route map showing the relationship between the transverse tangent vector and the reference difference vector in the third embodiment of the present invention.
FIG. 6 is a route map showing a conventional patch. S (Il+W)...Patch, P. . 1,
P... 1, PBz>, P nx+ "'"" Node S
P + Ill % P no % P (311%P
(321, P(+u, P T!0), P(131, P
(t3.””...Control point, P(,,,P(Iゎ,P□.,P,tゎ...Internal control point, BODI-B
OD4...Boundary curve, ao@5a02, a!
@Sa! ! , b00% b36Sb03, b3!・・・
...Control side vector, aoia. zsaz a!
! , blo, bto, b13, b t 3 ”・”・Transverse tangent vector, AoiiBOK・・・Reference difference vector.

Claims (1)

[Claims] A free-form surface is created by forming a large number of framework spaces surrounded by boundary curves through framework processing, and by extending patches expressed by vector functions having parameters representing positions in the framework spaces. In the free-form surface creation method, an internal control point corresponding to each of the boundary curves is set by a transverse tangent vector to each boundary curve, and a boundary corresponding to the transverse tangent vector is set. A free-form surface creation method characterized by selecting a rate of change in a direction along a curve to a predetermined value to prevent the positions of internal control points from intersecting, overlapping, or being close to each other.